On multiscale methods in Petrov–Galerkin formulation

详细信息    查看全文

• 作者：Daniel Elfverson ; Victor Ginting ; Patrick Henning
• 关键词：35J15 ; 65N12 ; 65N30 ; 76S05
• 刊名：Numerische Mathematik
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：131
• 期：4
• 页码：643-682
• 全文大小：1,816 KB
• 参考文献：1.Abdulle, A.: On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul. 4(2), 447-59 (2005)MATH MathSciNet CrossRef
  2.Abdulle, A., E,W., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1-7 (2012)
  3.Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482-518 (1992)MATH MathSciNet CrossRef
  4.Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers, London (1997)
  5.Babuska, I., Lipton, R.: Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9(1), 373-06 (2011)MATH MathSciNet CrossRef
  6.Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36(153), 35-1 (1981)MATH MathSciNet CrossRef
  7.Bank, R.E., Yserentant, H.: On the \(H^1\) -stability of the \(L_2\) -projection onto finite element spaces. Numer. Math. 126(2), 361-81 (2014)MATH MathSciNet CrossRef
  8.Bush, L., Ginting, V., Presho, M.: Application of a conservative, generalized multiscale finite element method to flow models. J. Comput. Appl. Math. 260, 395-09 (2014)MATH MathSciNet CrossRef
  9.Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN. Math. Model. Numer. Anal. 33(6), 1187-202 (1999)MATH MathSciNet CrossRef
  10.Carstensen, C., Verfürth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36(5), 1571-587 (1999)MATH MathSciNet CrossRef
  11.Efendiev, Y., Ginting, V., Hou, T., Ewing, R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155-74 (2006)MATH MathSciNet CrossRef
  12.Efendiev, Y., Hou, T., Ginting, V.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2(4), 553-89 (2004)MATH MathSciNet CrossRef
  13.Elfverson, D., Georgoulis, E.H., M?lqvist, A.: An adaptive discontinuous Galerkin multiscale method for elliptic problems. Multiscale Model. Simul. 11(3), 747-65 (2013)MATH MathSciNet CrossRef
  14.Elfverson, D., Georgoulis, E.H., M?lqvist, A., Peterseim, D.: Convergence of a discontinuous Galerkin multiscale method. SIAM J. Numer. Anal. 51(6), 3351-372 (2013)MATH MathSciNet CrossRef
  15.Gaspoz, F.D., Heine, C.-J., Siebert, K.G.: Optimal grading of the newest vertex bisection and \({H}^1\) -stability of the \({L}^2\) -projection. SimTech Universit?t, Stuttgart (2014)
  16.Ginting, V.: Analysis of two-scale finite volume element method for elliptic problem. J. Numer. Math. 12(2), 119-41 (2004)MATH MathSciNet CrossRef
  17.Gloria, A.: An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul. 5(3), 996-043 (2006)MATH MathSciNet CrossRef
  18.Gloria, A.: Reduction of the resonance error-part 1: approximation of homogenized coefficients. Math. Models Methods Appl. Sci. 21(8), 1601-630 (2011)MATH MathSciNet CrossRef
  19.Henning, P., M?lqvist, A.: Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36(4), A1609–A1634 (2014)MATH CrossRef
  20.Henning, P., M?lqvist, A., Peterseim, D.: A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM Math. Model. Numer. Anal. 48(5), 1331-349 (2014)MATH MathSciNet CrossRef
  21.Henning, P., M?lqvist, A., Peterseim, D.: Two-level discretization techniques for ground state computations of Bose–Einstein condensates. SIAM J. Numer. Anal. 52(4), 1525-550 (2014)MATH MathSciNet CrossRef
  22.Henning, P., Morgenstern, P., Peterseim, D.: Multiscale partition of unity. In: Meshfree Methods for Partial Differential Equations VII. Lecture Notes in Computational Science and Engineering, vol. 100. Springer, Berlin (2015)
  23.Henning, P., Ohlberger, M.: The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math. 113(4), 601-29 (2009)MATH MathSciNet CrossRef
  24.Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11(4), 1149-175 (2013)MATH MathSciNet CrossRef
  25.Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169-89 (1997)MATH MathSciNet CrossRef
  26.Hou, T.Y., Wu, X.-H., Zhang, Y.: Removing the cell resonance error in the multiscale finite element method via a Petrov–Galerkin formulation. Commun. Math. Sci. 2(2), 185-05 (2004)MATH MathSciNet CrossRef
  27.Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1-), 387-01 (1995)MATH CrossRef
  28.Hughes,
• 作者单位：Daniel Elfverson (1)
  Victor Ginting (2)
  Patrick Henning (3) (4)
  
  1. Department of Information Technology, Uppsala University, Box 337, 751 05, Uppsala, Sweden
  2. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA
  3. Section de Mathématiques, école polytechnique fédérale de Lausanne, 1015, Lausanne, Switzerland
  4. University of Münster, 48149, Münster, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

In this work we investigate the advantages of multiscale methods in Petrov–Galerkin (PG) formulation in a general framework. The framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space, which only contains negligible fine scale information. The multiscale space can then be used to obtain accurate Galerkin approximations. As a model problem we consider the Poisson equation. We prove that a Petrov–Galerkin formulation does not suffer from a significant loss of accuracy, and still preserve the convergence order of the original multiscale method. We also prove inf-sup stability of a PG continuous and a discontinuous Galerkin finite element multiscale method. Furthermore, we demonstrate that the Petrov–Galerkin method can decrease the computational complexity significantly, allowing for more efficient solution algorithms. As another application of the framework, we show how the Petrov–Galerkin framework can be used to construct a locally mass conservative solver for two-phase flow simulation that employs the Buckley–Leverett equation. To achieve this, we couple a PG discontinuous Galerkin finite element method with an upwind scheme for a hyperbolic conservation law. Mathematics Subject Classification 35J15 65N12 65N30 76S05